= Beam compliance from two support spans
{title2=$a=(C_2-C_1)/(L_2^3-L_1^3)$}
\b[Beam compliance from two support spans] separates a cubic span-dependent beam response from a span-independent contribution to the measured <mechanical compliance>. Assume the same uniform specimen, unchanged machine contribution $C_0$ and a small-deflection central-load model:
$$
C_i=C_0+aL_i^3,\qquad a=\frac1{48B},\qquad
a=\frac{C_2-C_1}{L_2^3-L_1^3}.
$$
Here $B$ is <beam flexural rigidity> and $L_1\ne L_2$. The <midspan compliance of a simply supported beam> contains the basic numerical fit, while the beam source derives the cubic dependence.<Beam Displacements by David Roylance>[Introduction; Example1, equation7]{ref}
An original uncertainty example shows why distinct spans can still be too close. Let the actual model have $B=10\,\mathrm{N\,m^2}$, hence $a=1/480\,\mathrm{N^{-1}m^{-2}}$. Assign independent compliance <standard uncertainties> $u(C_1)=u(C_2)=1\,\mathrm{\mu m/N}$, and neglect span uncertainty for this example. Since the slope estimate is linear in the readings,
$$
u(a)=\frac{\sqrt{u(C_1)^2+u(C_2)^2}}{|L_2^3-L_1^3|}.
$$
Spans $0.200$ and $0.210\,\mathrm m$ give $u(a)\approx0.00112\,\mathrm{N^{-1}m^{-2}}$, about54percent of the actual coefficient. Using $0.200$ and $0.300\,\mathrm m$ instead gives about3.57percent with the same readout uncertainties. The close-span subtraction strongly amplifies those errors. At the larger relative uncertainty, simply treating $B=1/(48a)$ as approximately linear is unreliable.
This <conditioning of two-span beam compliance inference> concerns measurement precision, separately from model adequacy. If shear adds a term proportional to $L$, then $C(L)=C_0+aL^3+bL$ has three unknown coefficients that two readings cannot identify. Changing contact or fixture compliance can likewise violate the assumed constant $C_0$. Further spans and an independent deflection readout test those assumptions; two fitted points alone do not.
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